from Superfluid Helium to Atomic Bose-Einstein Condensates Makoto TSUBOTA Department of Physics, Osaka City University, Japan

Transcription

1 from Superfluid Helium to Atomic Bose-Einstein Condensates Makoto TSUBOTA Department of Physics, Osaka City University, Japan Thanks to the many collaborators Review article M. Tsubota, J. Phys. Soc. Jpn.77 (2008) Progress in Low Temperature Physics Vol.16, eds. W. P. Halperin and M. Tsubota, Elsevier, 2008

2 My messages 1. Quantum turbulence (QT) has recently become one of the most important branches in low temperature physics. 2. QT has been studied thoroughly in superfluid 4 He and 3 He, and recently even in atomic Bose-Einstein condensates. 3. The research of QT can make a major breakthrough in the great mystery in nature, namely turbulence.

3 Leonardo Da Vinci ( ) Da Vinci observed turbulent flow and found that turbulence consists of many vortices. Turbulence is not a simple disordered state but having some structures with vortices.

4 Certainly turbulence looks to have many vortices. Turbulence behind a dragonfly However, these vortices are unstable; they repeatedly appear, diffuse and disappear. It is not so straightforward to confirm the Da Vinci message in classical turbulence.

5 Key concept of my talk The Da Vinci message is actually realized in quantum turbulence comprised of quantized vortices.

6 A quantized vortex is a vortex of superflow in a BEC. Any rotational motion in superfluid is sustained by quantized vortices. (i) The circulation is quantized. # v s! ds ="n ( n = 0,1, 2,L )! = h / m A vortex with n is unstable. Every vortex has the same circulation. (ii) Free from the decay mechanism of the viscous diffusion of the vorticity. The vortex is stable. (iii) The core size is very small. ρ s (r) The order of the coherence length. rot v s r

7 Classical Turbulence (CT) vs. Quantum Turbulence (QT) Classical turbulence Quantum turbulence Motion of vortex cores QT is much simpler The vortices are unstable. Not easy to identify each vortex. than CT, because each element of turbulence The circulation differs from one to another, not conserved. is definite. The quantized vortices are stable topological defects. Every vortex has the same circulation. Circulation is conserved.

8 There are two main cooperative phenomena of quantized vortices; Vortex array under rotation and Vortex tangle (Quantum turbulence). Superfluid He Vortex array Vortex tangle 2. Atomic BEC None 1. 3.

9 1. Quantized vortices in atomic BECs M.Tsubota, K.Kasamatsu, M.Ueda, Phys.Rev.A65, (2002) K.Kasamatsu, T.Tsubota, M.Ueda, Phys.Rev.A67, (2003)

10 Advantages of atomic BECs as a physical system Theoretical At low temperatures the model of dilute Bose gas and the mean field approximation theory (Gross-Pataevskii(GP) model) are quantitatively correct. ih "#(r,t) "t Experimental & = $ h2 % 2 2m + V 2) ( ext (r) + g #(r,t) + #(r,t) ' * We can directly control and visualize the condensate. We can change the atomic interaction by Feshbach resonance.

11 What happens if we rotate a vessel having a usual viscous classical fluid inside? The fluid rotates with the same angular velocity with the vessel. This means that there appears one vortex in the vessel. The single vortex can make the solid-body rotation with any angular velocity. This does not occur in quantum fluids! Such experiments were done in atomic BECs.

12 Observation of quantized vortices in atomic BECs ENS K.W.Madison, et.al PRL 84, 806 (2000) JILA P. Engels, et.al PRL 87, (2001) MIT J.R. Abo-Shaeer, et.al Science 292, 476 (2001) Oxford E. Hodby, et.al PRL 88, (2002)

13 How can we rotate the trapped BEC K.W.Madison et.al Phys.Rev Lett 84, 806 (2000) x z y 100µm Total potential cigar-shape Non-axisymmetric potential Rotation frequency Ω Optical spoon 5µm Axisymmetric potential V ext (R) = V trap (R) + U stir (R) U stir (R) = m 2 " 2 #($ x X 2 + $ y Y 2 ) " x # " y 20µm 16µm

14 Direct observation of the vortex lattice formation Snapshots of the BEC after turning on the rotation K.W. Madison et al. PRL 86, 4443 (2001)! = " R x 2 # R y 2 R x 2 + R y 2 Rx Ry 1. The BEC becomes elliptic, then oscillating. 2. The surface becomes unstable. 3. Vortices enter the BEC from the surface. 4. The BEC recovers the axisymmetry, the vortices forming a lattice.

15 The Gross-Pitaevskii(GP) equation in a rotating frame Wave function ih "# "t ih "# "t Two-dimensional "(r,t) = $ h2 2m % 2 # + V trap # + g # 2 # in a rotating frame Interaction g = 4"h2 a s m = $ h2 2m % 2 # + (V trap + U stir )# + g # 2 # $ &L z # U stir (R) = m 2 " 2 #($ x X 2 + $ y Y 2 ) a s s-wave scattering length Ω simplified Ω

16 The GP equation with a dissipative term ih "# "t ' = $ h2 2m % * ) 2 + (V trap + U stir ) + g # 2 $ µ $ &L z,# ( + (i " #)h $% $t " = 0.03 : dimensionless parameter S.Choi, et.al. PRA 57, 4057 (1998) I.Aranson, et.al. PRB 54, (1996) This dissipation comes microscopically from the interaction between the condensate and the noncondensate. E.Zaremba, T. Nikuni, and A. Griffin, J. Low Temp. Phys. 116, 277 (1999) C.W. Gardiner, J.R. Anglin, and T.I.A. Fudge, J. Phys. B 35, 1555 (2002) M. Kobayashi and M. Tsubota, PRL 97, (2006)

17 Profile of a single quantized vortex h2 2 2! " # + Vtrap# + g # # = µ# 2m A quantized vortex!(r ) = n0 (r)e i"( r ) A vortex n ! Velocity field! r Vortex core= healing length h "# 2mgn 0

18 Dynamics of the vortex lattice formation (1) Time development of the condensate density n 0 " = 0.7# $ Tsubota et al., Phys. Rev. A 65, (2002) Experiment V trap (r) = 1 2 m! 2 " r 2!(r) = n 0 (r)e i"( r ) K.W.Madison et al. PRL 86, 4443 (2001)

19 Dynamics of the vortex lattice formation (2) Time-development of the condensate density n 0 t=0 67ms 340ms 390ms 410ms 700ms Are these holes actually quantized vortices?

20 Dynamics of the vortex lattice formation (3) Time-development of the phase!!(r) = n 0 (r)e i"( r )

21 Dynamics of the vortex lattice formation (4) Time-development of the phase! t=0 67ms 340ms Ghost vortices Becoming 390ms real vortices 410ms 700ms

22 Dynamics of the vortex lattice formation (5)!/" Rx Ry! = " R 2 2 x # R y R 2 2 x + R y time (msec) K.W.Madison et.al. Phys. Rev. Lett. 86, 4443 (2001)

23 Simultaneous display of the density and the phase

24 The three-dimensional dynamics K.Kasamatsu, M. Machida, N. Sasa, M.Tsubota, PRA 71, (2005) Space resolution γ=0.03 The dynamics is qualitatively similar to the two-dimensional case, but we have Excitation and motion of Kelvin waves Turbulence made of phase defects Density Superflow

25 There are two main cooperative phenomena of quantized vortices; Vortex array under rotation and Vortex tangle (Quantum turbulence). Superfluid He Vortex array Vortex tangle 2. Atomic BEC None 1. 3.

26 2. Quantum turbulence in superfluid helium Liquid 4 He enters the superfluid state at 2.17K (λ point) with Bose-Einstein condensation. Its hydrodynamics is well described by the two-fluid model. The two-fluid model The system is a mixture of inviscid superfluid and viscous normal fluid.! =! s +! n j = " s v s + " n v n point Temperature density velocity viscosity entropy Superfluid Normal fluid " s ( T) " n ( T) v s ( r) v n 0 0 ( r) " n ( T) s n ( T)

27 The two-fluid model could explain various phenomena of superfluidity which were observed experimentally. Thermo-mechanical effect, film flow, etc. However,.

28 The superfluidity breaks down when it flows fast. (i) v s < vc v s (some critical velocity) t! " v s There is no interaction between two fluids, and the superfluid can flow forever without decaying. (ii) v > v s c v s v =0 s A tangle of quantized vortices grows. The two fluids interact through the mutual friction due to the tangle, and the superflow decays.

29 1955 Feynman proposed that superfluid turbulence consists of a tangle of quantized vortices. Progress in Low Temperature Physics Vol.I (1955), p Hall and Vinen observed superfluid turbulence. The mutual friction between the vortex tangle and the normal fluid causes the dissipation of the flow.

30 Lots of experimental studies were done chiefly for thermal counterflow of superfluid 4 He. Vortex tangle Heater Normal flow Super flow 1980 s K. W. Schwarz Phys.Rev.B38, 2398(1988) Made the direct numerical simulation of the three-dimensional dynamics of quantized vortices and succeeded in explaining quantitatively the observed temperature difference.

31 Development of a vortex tangle in a thermal counterflow Schwarz, Phys.Rev.B38, 2398(1988). Schwarz obtained numerically the statistically steady state of a vortex tangle which is sustained by the competition between the applied flow and the mutual friction. The obtained vortex density L(v ns, T) agreed quantitatively with experimental data. v s v n The research of counterflow turbulence has been successful.

32 Most studies of superfluid turbulence were devoted to thermal counterflow. No analogy with classical turbulence When Feynman showed the above figure, he thought of a cascade process in classical turbulence. What is the relation between superfluid turbulence and classical turbulence

33 How can we study the similarity and the difference between QT and CT? Let s focus on the most important statistical law in CT, namely, the Kolmogorov law. E(k) = C" 2 / 3 k #5 / 3

34 Energy spectra of fully developed turbulence Energy spectrum of the velocity field E = 1 2 " " 2 v dr = E(k)dk Energy-containing range Energycontaining range Inertial range Kolmogorov law Energy-dissipative range Energy spectrum of turbulence The energy Richardson is injected into cascade the system process at k! k 0 = 1/ l 0. Inertial range Dissipation does not work. The nonlinear interaction transfers the energy from low k region to high k region. Kolmogorov law : E(k)=Cε 2/3 k -5/3 Energy-dissipative range The energy is dissipated with the rate ε at the Kolmogorov wave number k c = (ε/ν 3 ) 1/4.

35 Everybody believes that turbulence is sustained by this Richardson cascade. However, this is only a cartoon; nobody has ever confirmed it clearly. One of the reasons is that it is too difficult to identify each eddy in a fluid.

36 Quantum turbulence may give a prototype of turbulence, much simpler than conventional turbulence. Can such quantized vortices still produce the essence of turbulence, for example, the Kolmogorov law? Having this sort of motivation, the studies of QT have entered a new stage since the middle of 90 s s!

37 New study of quantum turbulence (1) Maurer and Tabeling, Europhysics. Letters. 43, 29(1998) 1.4K < T < T λ Measurements of local pressure in flows driven by two counterrotating disks finds the Kolmogorov spectrum. Energy spectra Stalp, Skrbek and Donnely, Phys.Rev.Lett. 82, 4831(1999) 1.4 < T < 2.15K Decay of grid turbulence. The data of the second sound attenuation was consistent with a classical model with the Kolmogorov spectrum.

38 New study of quantum turbulence (2) Vinen, Phys.Rev.B61, 1410(2000) Considering the relation between ST and CT The Oregon s result is understood by the coupled dynamics of the superfluid and the normal fluid due to the mutual friction. the characteristic vortex spacing in a tangle. Length scales are important, compared with Kivotides, Vassilicos, Samuels and Barenghi, Europhysics Lett. 57, 845(2002) When superfluid is coupled with the normal-fluid turbulence that obeys the Kolmogorov law, its spectrum follows the Kolmogorov law too. What happens at very low temperatures? Is there still the similarity or not?

39 Energy spectra of quantum turbulence (QT) There are three works which directly study the energy spectrum of QT at zero temperature. Decaying Kolmogorov turbulence in a model of superflow C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644 (1997) The Gross-Pitaevskii (GP) model Energy Spectrum of Superfluid Turbulence with No Normal-Fluid Component T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, (2002) The vortex-filament model Kolmogorov Spectrum of Superfluid Turbulence: Numerical Analysis of the Gross-Pitaevskii Equation with a Small-Scale Dissipation M. Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, (2005), J. Phys. Soc. Jpn.74, 3248 (2005).

40 C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) t By using the GP model, they obtained a vortex tangle with starting from the Taylor- Green vortices. We should note that the GP model describes a compressible fluid.

41 In order to study the Kolmogorov spectrum, it is necessary to decompose the total energy into some components. (Nore et al., 1997) Total energy E = 1 " dx! dx# * & $% 2 + g 2 # 2 ) " '( * + # The kinetic energy the compressible part and E = E int + E q + E kin! = " exp( i# ) E kin = " E kin c = 1 dx! " 1 " dx! ( ) dx # $% " is divided into with the incompressible part i 1 E kin = with. " dx! " dx ( # $% ) i 2 div! "# 2 [ ] dx ( # $% ) c 2 [ ] rot (! "# ) c = 0 This incompressible kinetic energy E kin i should obey the Kolmogorov spectrum. ( ) i = 0

42 C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) 2 < k < 12 2 < k < 14 2 < k < 16 E(k) k -n(t) n(t) E(k) 5/3 t The right figure shows the energy spectrum at a moment. The left figure shows the development of the exponent n(t). The exponent n(t) goes through 5/3 on the way of the dynamics. In the late stage, however, the exponent deviates from 5/3, because the sound waves resulting from vortex reconnections disturb the cascade process of the inertial range. k

43 Reconnection of quantized vortices (Analysis of the Gross-Pitaevskii equation)

44 C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) 2 < k < 12 2 < k < 14 2 < k < 16 E(k) k -n(t) n(t) E(k) 5/3 t The right figure shows the energy spectrum at a moment. The left figure shows the development of the exponent n(t). The exponent n(t) goes through 5/3 on the way of the dynamics. In the late stage, however, the exponent deviates from 5/3, because the sound waves resulting from vortex reconnections disturb the cascade process of the inertial range. k

45 Kolmogorov spectrum of quantum turbulence M. Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, (2005), J. Phys. Soc. Jpn. 74, 3248 (2005) 1. We solved the GP equation in the wave number space in order to use the fast Fourier transformation. 2. We made a steady state of turbulence. In order to do that, 2-1 We introduced a dissipative term which dissipates the Fourier component of the high wave number, namely, phonons of short wave length. 2-2 We excited the system at a large scale by moving a random potential.

46 The GP equation in the real space i!! t " r, t [ ]" r, t ( ) = #$2 # µ + g "( r, t) 2 ( ) The GP equation in the Fourier space i!!t " k,t ( ) = k 2 # µ! 2 = 1 g " 2 ( ) ( ) " k, t + g " k V 2 1,t $ k 1,k 2 ( ) " * k 2,t ( )" k # k 1 + k 2,t ( ) healing length giving the vortex core size To solve the GP equation numerically with high accuracy, we use the Fourier spectral method in space with the periodic boundary condition in a cube.

47 How to dissipate the energy at small scales? The GP equation with the small scale dissipation {i!" (k)} # #t $ ( k,t) = ( k2! µ ) $ ( k,t)! 2 = 1 g " 2! (k ) =! 0 "( k # 2$ / %) + g $ k V 2 1,t % k 1,k 2 ( ) $ * k 2,t ( )$ k! k 1 + k 2,t ( ) :healing length giving the vortex core size We introduce the dissipation that works only in the scale smaller than ξ. Since there is no vortex motion at the scales smaller than ξ, this dissipation must work for only short-wavelength sound waves. cf. M. Kobayashi and M. Tsubota, PRL 97, (2006)

48 How to inject the energy at large scales? This is done by moving the random potential satisfying the space-time correlation: # V(x, t)v( x!, t!) = V 2 0 exp " ( x " x!)2 2 " ( t " t!)2 & % 2 ( $ % 2X 0 2T 0 '( The variable X 0 determines the scale of the energy-containing range. V 0 =50, X 0 =4 and T 0 = X 0 ξ

49 Thus steady turbulence is obtained.(1) Time development of each energy component Vortices Phase in a central plane Moving random potential

50 Thus steady turbulence is obtained.(2) Time development of each energy component

51 Energy spectrum of the steady turbulence The energy spectrum obeys the Kolmogorov form. Quantum turbulence is found to show the essence of classical turbulence! The inertial range is sustained by the genuine Richardson cascade of quantized vortices. 2π / X 0 2π / ξ M. Kobayashi and M. Tsubota, J. Phys. Soc. Jpn. 74, 3248 (2005)

52 3. Quantum turbulence in a trapped Bose-Einstein condensate M. Kobayashi and M. Tsubota, Phys. Rev. A76, (2007) Two main cooperative phenomena of quantized vortices are Vortex array and Vortex tangle. Superfluid He Vortex array Vortex tangle Atomic BEC None

53 Is it possible to make turbulence in a trapped BEC? (1)We cannot apply some flow to the system. (2) This is a finite-size system. Can we make turbulence with enough inertial range? Then the coherence length is not much smaller than the system size. However, we could confirm the Kolmogorov law.

54 How to make turbulence in a trapped BEC z 1. Trap the BEC in a weakly elliptic potential. U( x) = m " 2 2 [( 1#$ 1 )( 1#$ 2 )x 2 + ( 1+ $ 1 )( 1#$ 2 )y 2 + ( 1+ $ 2 )z 2 ] x y 2. Rotate the system first around the x-axis, next around the z-axis. "( t) = (" x, " z sin" x t, " z cos" x t)

55 Actually this idea has been already used in CT. S. Goto, N. Ishii, S. Kida, and M. Nishioka, Phys. Fluids 19, (2007) Rotation around one axis Rotation around two axes

56 Are these two rotations represented by their sum? No! Precession Spin axis itself rotates around another axis. Ω x Precessing motion of a gyroscope Ω z We consider the case where the spinning and precessing rotational axes are perpendicular to each other. Hence, the two rotations do not commute, and thus cannot be represented by their sum.

57 Two precessions (ω x ω z ) Three precessions (ω y ω x ω z ) Condensate density Quantized vortices Condensate density Quantized vortices

58 Rotation around two axes and three axes -comparison of the energy spectra- Rotation around two axes Rotation around three axes Rotation around three axes makes more isotropic QT whose η is closer to 5/3.

59 What can we learn from QT of atomic BEC? Controlling the transition to turbulence by changing the rotational frequency or interaction parameters, etc. We can visualize quantized vortices. We can consider the relation of real space cascade of vortices and the wavenumber space cascade (Kolmogorov s law). Changing the trapping potential or the rotational frequency leads to dimensional crossover (2D D) in turbulence.

60 Controlling the transition to turbulence Ω z Vortex tangle ω x ω z Vortex lattice? 0 Vortex lattice Ω x

61 What can we learn from quantum turbulence of atomic BEC? Controlling the transition to turbulence by changing the rotational frequency or interaction parameters, etc. We can visualize quantized vortices. We can consider the relation of real space cascade of vortices and the wavenumber space cascade (Kolmogorov s law). Changing the trapping potential or the rotational frequency leads to dimensional crossover (2D D) in turbulence. z z x y y x x z y

62 Very recently quantum turbulence was realized also in atomic BECs! Henn et al., PRL103, (2009) Coupled large amplitude oscillation

63 Summary 16th century We discussed the recent interests in quantum turbulence. Quantum turbulence consists of quantized vortices which are stable topological defects. Quantum turbulence may give a prototype of turbulence much simpler than conventional turbulence. 21st century

64 Rotating superfluid and the vortex lattice " = n 0 e i# Ω < Ωc Ω > Ωc F = E " #L z Minimizing the free energy in a rotating frame. The triangular vortex lattice sustains the solid body rotation. Yarmchuck and Packard v s = h m "# " # v s = 0 Vortex lattice observed in rotating superfluid helium

65 Comments 4-1. Do we need dissipation for the vortex lattice formation in a rotating BEC? 4-2. Full Biot-Savart calculation for thermal counterflow

66 4-1. Do we need dissipation for the vortex lattice formation in a rotating BEC? M. Kobayashi, K. Kasamatsu, M. Ueda, M. Tsubota, in preparation. There is some controversy about this problem. We can conclude that we do need dissipation. ih "# "t ' = $ h2 2m % * ) 2 + (V trap + U stir ) + g # 2 $ µ $ &L z,# ( +

67 C. Lobo, A. Sinatra, Y. Castin, Phys. Rev. Lett. 92, (2004) The authors say that a vortex lattice is formed even without dissipation. 3D simulation Grid size

68 2D Simulation without dissipation for different mesh sizes The dynamics depends strongly on the mesh size!

69 The finer the mesh becomes, the longer it takes for the vortex lattice to form. Angular momentum At τ, the vortices enter the condensate. τ becomes infinite for Δx 0. Vortices never enter the condensate for infinitely small mesh!

70 Therefore, the statement a vortex lattice is formed even without dissipation is doubtful. Conclusion The system needs some dissipation in order to form a vortex lattice.

71 4-2. Full Biot-Savart calculation for thermal counterflow Makoto Tsubota, Hiroyuki Adachi Osaka City university, Japan

72 Vortex filament model (Schwarz) s r A vortex makes the superflow of the Biot-Savart law, and moves with this local flow. At a finite temperature, the mutual friction should be considered. s 0 = " 4# s $ % s $ + & 4# s = s 0 + ) s $ % v n ' s 0 ( ) ' $ ( s 1 ' r) % ds ( ' 1 + v L s 1 ' r 3 s,a s ) s $ % s $ % v n ' s 0 ( ) [ ( )] The approximation neglecting the nonlocal term is called the LIA(Localized Induction Approximation). s 0 = " 4# s $ % s $ + v s,a s ( )

73 Schwarz s simulation(1) PRB38, 2398(1988) Schwarz simulated the counterflow turbulence by the vortex filament model and obtained the steady state. However, this simulation was unsatisfactory. 1. All calculations were performed by the LIA.

74 Schwarz s simulation(2) PRB38, 2398(1988) However, this simulation was unsatisfactory. 1. All calculation was performed by the LIA. 2. He used an artificial mixing procedure in order to obtain the steady state.

75 After Schwarz, there has been no progress on the counterflow simulation. In this work we made the steady state of counterflow turbulence by fully nonlocal simulation. An important criterion of the steady state is to obtain

76 Simulation by the full Biot-Savart law BOX (0.1cm) 3 T=1.6(K) Vns=0.367cm/s Periodic boundary conditions for all three directions

77 Comparison between LIA and full Biot-Savart Full Biot-Savart LIA Vortices become anisotropic, forming layer structures.

78 Developments of the line-length density between LIA and Full Biot-Savart,)#-./ $ + $"!! $!!! #"!! #!!! "!!! LIA Full Biot-Savart! #! $! %! &! "! '()*+ T=1.6 K Vns=0.367 cm/s box=(0.2 cm) 3

79 Anisotropic parameter 122-.,0!"'$!"'!"&$!"&!"%$!"%!"#$!"# LIA Full Biot-Savart! (! )! *! +! $!,-./0 T=1.6 K Vns=0.367 cm/s box=(0.2 cm) 3

80 L vs. Vns by full Biot-Savart simulation T=1.9K box (0.1cm) 3! "#$ 3"#45 $ /0 $0! "#$ %&%"'()(*%+,-%.%")/''* We first obtained a statistically steady state by full Biot-Savart simulation )$ 0)/ 0)1 0)2 +,-%345#-6 Numerical results Experiment